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Section 5.7 : Combining Functions
Learning Targets: F.BF.1.b
Important Terms and Definitions
We are able to take all of the knowledge we have about
arithmetic operations and apply it to functions. Functions can be
added, subtracted, multiplied, and divided.
Addition: (𝑓 + 𝑔)(𝑥) = 𝑓(𝑥) + 𝑔(𝑥)
Subtraction: (𝑓 − 𝑔)(𝑥) = 𝑓(𝑥) − 𝑔(𝑥)
Multiplication: (𝑓 ∙ 𝑔)(𝑥) = 𝑓(𝑥) ∙ 𝑔(𝑥)
Division: (𝑓𝑔
) (𝑥) = 𝑓(𝑥)𝑔(𝑥)
, 𝑔(𝑥) ≠ 0
Example: Given that 𝑓(𝑥) = 3𝑥 + 4 and 𝑔(𝑥) = 2𝑥 − 3, find (𝑓 +
𝑔)(𝑥), (𝑓 − 𝑔)(𝑥), (𝑓 ∙ 𝑔)(𝑥), and (𝑓
𝑔) (𝑥).
(𝑓 + 𝑔)(𝑥) = 𝑓(𝑥) + 𝑔(𝑥) = 3𝑥 + 4 + 2𝑥 − 3 = 5𝑥 + 1
(𝑓 − 𝑔)(𝑥) = 𝑓(𝑥) − 𝑔(𝑥) = 3𝑥 + 4 − (2𝑥 − 3) = 3𝑥 + 4 − 2𝑥 + 3 =
𝑥 + 7
(𝑓 ∙ 𝑔)(𝑥) = 𝑓(𝑥) ∙ 𝑔(𝑥) = (3𝑥 + 4)(2𝑥 − 3) = 6𝑥2 − 9𝑥 + 6𝑥 −
12
= 6𝑥2 − 3𝑥 − 12
(𝑓𝑔
) (𝑥) = 𝑓(𝑥)𝑔(𝑥)
= 3𝑥+42𝑥−3
(ex 1) Given that 𝑓(𝑥) = 6𝑥 − 1 and 𝑔(𝑥) = 3𝑥, find (𝑓 + 𝑔)(𝑥),
(𝑓 − 𝑔)(𝑥), (𝑓 ∙ 𝑔)(𝑥), and (𝑓
𝑔) (𝑥).
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(ex 2) Given that 𝑓(𝑥) = 144𝑥 − 48 and 𝑔(𝑥) = 12, find (𝑓 +
𝑔)(𝑥), (𝑓 − 𝑔)(𝑥), (𝑓 ∙ 𝑔)(𝑥), and (𝑓
𝑔) (𝑥).
Finding a Combination of Functions at a Given Value
Example: Given that 𝑓(𝑥) = 3𝑥 + 4 and 𝑔(𝑥) = 2𝑥 − 3, find (𝑓 +
𝑔)(3).
(𝑓 + 𝑔)(3) = 𝑓(3) + 𝑔(3) = 3(3) + 4 + 2(3) − 3 = 9 + 4 + 6 − 3 =
16
(ex 3) Given that 𝑓(𝑥) = 3 ∙ 8𝑥 and 𝑔(𝑥) = 2 ∙ 8𝑥 + 1, find (𝑓 +
𝑔)(2) and (𝑓 − 𝑔)(3).
(ex 4) Given that 𝑓(𝑥) = 5 ∙ 3𝑥 and 𝑔(𝑥) = 2 ∙ 3𝑥 + 6, find (𝑓 ∙
𝑔)(2) and (𝑓𝑔
) (3).
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(ex 5) Scientists conducted a study of the sparrow populations
in Boardman and Canfield. The results of the study projected that
the population in Boardman in x years will be modeled by 𝐵(𝑥) = 950
∙ 1.29𝑥 − 120𝑥 and the population in Canfield in x years will be
modeled by 𝐶(𝑥) = 500𝑥 − 600. What is a function giving the
difference in population of sparrows between Boardman and Canfield?
Predict what the difference in population will be in 6 years.
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Homework – Section 5.7 : Combining Functions
Find (𝑓 + 𝑔)(𝑥), (𝑓 − 𝑔)(𝑥), (𝑓 ∙ 𝑔)(𝑥), and (𝑓𝑔) (𝑥) for the
following pairs of functions.
1. 𝑓(𝑥) = 15𝑥 + 3, 𝑔(𝑥) = 3𝑥 2. 𝑓(𝑥) = 2𝑥, 𝑔(𝑥) = 5𝑥 + 2 3. 𝑓(𝑥)
= 3𝑥 + 2, 𝑔(𝑥) = 5𝑥 + 1
4. Given 𝑓(𝑥) = 9 ∙ 2𝑥 + 16𝑥 − 6, 𝑔(𝑥) = (−8) ∙ 2𝑥 − 17, find (𝑓
+ 𝑔)(𝑥), (𝑓 + 𝑔)(4), (𝑓 − 𝑔)(𝑥) and (𝑓 − 𝑔)(1).
5. Given 𝑓(𝑥) = 9𝑥 + 3 − 6𝑥, 𝑔(𝑥) = 3𝑥, find (𝑓 ∙ 𝑔)(𝑥), (𝑓 ∙
𝑔)(3), (𝑓𝑔) (𝑥) and
(𝑓𝑔) (4).
Write a pair of functions that have the following
characteristics.
6. (𝑓 − 𝑔)(𝑥) = 2𝑥 + 3 7. (𝑓 ∙ 𝑔)(𝑥) = 12𝑥 + 8 8. (𝑓 + 𝑔)(𝑥) = 4
∙ 6𝑥 − 2
9. The cost of producing x baseball gloves is modeled by 𝐶(𝑥) =
20 + 15𝑥. The markup price of selling baseball gloves is modeled by
𝑀(𝑥) = 1.5. What is a function giving the selling price of the
gloves? How much would a coach spend if he wanted to buy gloves for
9 of his players?
10. The number of widgets produced in x days will be modeled by
the function 𝑊(𝑥) = 800 + 16𝑥 and the number of defective widgets
in x days will be modeled by 𝐷(𝑥) = 4 + 2𝑥. Write a function giving
the percent of defective widgets. Predict the percent of defective
widgets produced in 12 days. Round to the nearest tenth of a
percent.